Given

• Orders: , , ,

• Coefficient matrices:

• Intercept:
• Initial conditions: ,
• Two series stacked into :

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
	12	10	8	11	14	12	9	13	16	14	11	15	18	16	13	17

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
	8	9	7	6	10	11	9	8	12	13	11	10	14	15	13	12

Step 1 — formula

Substitute orders into the VARIMA recursion:

Difference — apply component-wise:

In differenced space, the model is VARMA:

Forecast the difference (set ):

Undifference — convert back to a forecast for :

Innovation:

Step 2 — apply at

First difference: .

Plug in , , , :

Step 3 — iterate

Pipeline at each : difference VARMA-forecast undifference. Values rounded to 4 decimal places.

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